Assume that $\textrm{C}$ and $\textrm{D}$ are categories and $F\colon \textrm{C}\to \textrm{D}$ and $G\colon \textrm{D}\to \textrm{C}$ adjoint functors, i.e. $F\dashv G$. My definition of adjunction is given in terms of the universal property: there exists a natural transformation (unit) $\eta\colon 1\to GF$ such that for all objects $X$ in $\textrm{C}$, for all objects $Y$ in $\textrm{D}$ and for all arrows $f\colon X\to G(Y)$ there exists a unique arrow $f'\colon F(X)\to Y$ such that $G(f') \ \eta_X =f$. Equivalently, we can reformulate it en terms of the counit $\epsilon$.
Now assume that $F\dashv G$ with fixed unit and counit $\eta$ and $\epsilon$. I want to prove that for all object $Y$ in $\textrm{D}$ we have $$G(\epsilon_Y)\circ\eta_{GY}=1_{GY}.$$
I am having trouble to prove that identity, using only the definition above... Can you give me a hint?